Bayesian Estimation for Hidden Information of a Single Molecular Motor

In single-molecule experiments of molecular motors, it is a widely adopted strategy to visualize the motion of a molecule by attaching a large probe particle. In such a situation, only the motion of the probe can be monitored while the motion of the molecule is hidden. Therefore, in order to study a stepwise motion of the motor in detail, one has to estimate the system parameters and the hidden trajectory of the motor molecule at the same time on the basis of the observed trajectory of the probe. In this presentation, we will present a novel approach for this problem, which is based on the statistical mechanics and empirical Bayesian estimation. The method consists of several steps. First, we temporarily fix the parameters of the system. Then, we can utilize the Bayes theorem to estimate the trajectory of the motor from the trajectory of the probe. On the basis of the estimated trajectory, the marginal likelihood can be calculated using the WKB method. Finally, by maximizing the marginal likelihood, we can estimate the system parameters along with the trajectory of the motor. Although the above mentioned tasks appear straightforward, it is necessary to adopt appropriate approximation schemes in order to reduce the computational cost as well as artifacts resulting from the finite time resolution of the measurements. It is found that our method gives a reasonable solution to this problem. We will discuss the effectiveness of the method presented here by using a simple model that consists of two Brownian particles (a molecule and a probe) connected by a spring.